Sequential Asset Voting is a tweak to Asset Voting that can be done several ways. One is by doing Asset sequentially in the multiwinner case: multiple rounds of negotiations are done, and after each round, the candidate(s) with the most votes are elected. It is also possible to spend or exhaust some votes after each seat is elected in Sequential Asset, which can influence the negotiation and potentially make Sequential Asset move in a continuum between majoritarianism and proportionality. If a Droop quota of votes is spent each time, for example, then Sequential Asset becomes at least semi-proportional based on negotiator preferences.
If only one candidate is elected at a time in Sequential Asset, with no reweighting/spending/exhausting of votes after each round, it becomes Bloc Asset Voting, which can allow a majority (as determined by negotiator preferences, not necessarily voter preferences) to win every seat. It is akin to deterministic Bloc methods such as Bloc Approval Voting.
It may be possible to do Bloc Asset Voting using regular Asset Voting rules, if when a candidate has more than (a majority of votes divided by number of seats to be filled) votes, they are required to either give any votes over that threshold away or forfeit them, and if a sequential tiebreaker is used where the negotiators who held a majority of votes at the start of the negotiations can decide who in each tie wins. As an example, here are the votes for a 3-winner election:
(1 A1, 1 A2, 1 A3, 1 A4) (1 B1, 1 B2, 1 B3)
This equates to 4 votes for the A faction and 3 for the B faction. In regular Asset, if the A and B factions act in their own interests, one possible final result is:
(2 votes each for two A candidates, 3 votes for one B candidate)
so that two A candidates and one B candidate will win. Neither faction can get a better result, because if the A faction tries to divide its votes evenly between three of its candidates to make all of them win, none of them will have more votes (1.333 votes each) than the B faction's preferred candidate (3 votes), so that the B faction will be able to force at least one of their candidates to win; similar reasoning applies to the B faction. But with Bloc Asset implemented as specified above, no candidate can have more than (3.5/3 = 1.167) votes, so one possible split in the final result would be:
(1.167 votes each for 3 A candidates, 1.167 votes each for 2 B candidates, 0.5 votes for 1 A candidate, and 0.666 votes for 1 B candidate)
3 A candidates tie with 2 B candidates, and using the sequential majoritarian tiebreaker, the A candidates originally had 4 votes to the 3 votes for the B candidates, so that the 3 A candidates can make themselves win each of their ties. The B faction, because they're prevented by the rules from coalescing more than 1.167 out of their 3 votes behind any of their given candidates, can't beat the A faction's split. This version of Bloc Asset essentially forces vote-splitting and vote-wasting to occur in the negotiations to prevent proportional results. If, say, the 3 B voters had all picked the same candidate before the negotiation, then these rules for Bloc Asset would require that 1.8333 of that B candidate's votes must be given to other candidates or would otherwise be wasted.
Note that it is also possible to make this implementation of Bloc Asset more proportional by modifying the threshold of votes a candidate may have; if, in the above example, any candidate could've had up to, say, 2.5 votes, then the final result could've looked like:
(2 votes each for two A candidates, 2.5 votes for one B candidate, and 0.5 votes for one B candidate)
Then 2 A candidates and one B candidate would've won, and the A faction wouldn't have been able to improve on their result.
Sequential Asset and Bloc Asset can also be done algorithmically. Bloc Asset's most relevant algorithmic version is essentially equivalent to sequentially electing the candidates at the top of a Smith-efficient Condorcet method's order of finish until all seats are filled, while Algorithmic Sequential Asset's most relevant algorithmic version can be treated as sequential Condorcet or Condorcet PR elections where ballots can, based on the rules, be reweighted after each round.